πn scattering and π photoproduction in a dynamical coupled-channels model

Transcription

1 π scattering and π photoproduction in a dynamical coupled-channels model Fei Huang Department of Physics and Astronomy, The University of Georgia, USA Collaborators: K. akayama (UGA), H. Haberzettl (GWU) M. Döring (Bonn U.), S. Krewald (FZ-Jülich) April 25, 211 JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

2 Outline Introduction Why s The current status of s Methodology for study Dynamical coupled-channels model Features of Jülich π hadron exchange model Partial wave amplitudes of π π ew developments: inclusion of KY channel & π photoproduction π + p Σ + K + cross sections & polarizations Improvements on π π amplitudes Gauge invariance Cross sections & spin asymmetries for π photoproduction Summary & perspectives JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

3 Why s. Isgur, Excited ucleons and Hadron Structure (STAR2), ewport ews, 2: There are three main reasons (one for each quark) why I believe that s deserve the special attention they get from the series of Workshops to which 2 belongs: The first is that nucleons are the stuff of which our world is made. As such they must be at the center of any discussion of why the world we actually experience has the character it does. I am convinced that completing this chapter in the history of science will be one of the most interesting and fruitful areas of physics for at least the next thirty years. My second reason is that they are the simplest system in which the quintessentially nonabelian character of QCD is manifest. There are, after all, c quarks in a proton because there are c colors. The third reason is that history has taught us that, while relatively simple, baryons are sufficiently complex to reveal physics hidden from us in the mesons. There are many experiences of this, but one famous example should suffice: Gell-Mann and Zweig were forced to the quarks by giving the octet and decuplet, while mesons admitted of many possible solutions. JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

4 s from Particle Data Book 21 Particle Status as seen in L 2I 2J Overall status π η ΛK ΣK π ρ γ (939) P11 (144) P11 (152) D13 (1535) S11 (165) S11 (1675) D15 (168) F15 (17) D13 (171) P11 (172) P13 (19) P13 (199) F17 (2) F15 (28) D13 (29) S11 (21) P11 (219) G17 (22) D15 (222) H19 (225) G19 (26) I1 11 (27) K113 (1232) P33 F (16) P33 o (162) S31 r (17) D33 b (175) P31 i (19) S31 d (195) F35 d (191) P31 e (192) P33 n (193) D35 (194) D33 F (195) F37 o (2) F35 r (215) S31 b (22) G37 i (23) H39 d (235) D35 d (239) F37 e (24) G39 n (242) H311 (275) I3 13 (295) K (well established) P 33(1232) P 11(144) D 13(152) S 11(1535) S 31(162) S 11(165) D 15(1675) F 15(168) D 33(17) P 13(172) F 35(195) P 31(191) F 37(195) G 17(219) H 19(222) G 19(225) H 311(242) 6 (no evidence from SAID) P 33(16) D 13(17) P 11(171) P 33(192) D 35(193) I 111(26) 12 (exist?) S 31(19) P 13(19) F 17(199) F 35(2) F 15(2) D 13(28) D 15(22) H 39(23) G 39(24) K 113(27) I 313(275) K 315(295) 8 (exist?) P 31(175) D 33(194) S 11(29) P 11(21) S 31(215) G 37(22) D 35(235) F 37(239) PWA s from Karlsruhe-Helsinki (Höhler 79) & Carnegie-Mellon (Cutkosky 8). Only 4-star s confirmed by GWU-SAID (Arndt 6). JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

5 Some specific questions of study Do the 1-star, 2-star & 3-star s exist? How many resonances in total? The symmetry of baryon spectrum? SU(6)? Where are the missing resonances? The structures of states? Excited qqq states or baryon-meson states? Mass reverse of S 11 (1535) and P 11 (144)? Quark-diquark configuration? Pentaquark? Dominant interactions among constituent quarks? OGE? GBE? Instanton? JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

6 Methodology for study complete set of data for γ KY There are lots of high precision data from JLab, MIT-Bates, BL-LEGS, Mainz-MAMI, Bonn-ELSA, GRAAL, Spring-8, et al. s are unstable and couple strongly to baryon-meson states Build coupled-channels meson-baryon reaction models to analyze the meson production data extract parameters understand the reaction mechanisms understand the structures and dynamical origins of Most widely used models: K matrix approximation, chiral unitary approach, dynamical coupled-channels model, et al. JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

7 Theoretical approaches T = V + V T T = V + V G T T: amplitude V: pseudo-potential G : free propagator Sum over all channels and partial waves, and integrate over intermediate momenta. 1 K matrix approximation G = E H +iǫ = P iπδ(e H E H ) iπδ(e H ) Principle-value part of the loop integral neglected off-shell intermediate states omitted integral equation reduced to algebraic equation Unitarity satisfied but analyticity violated Chiral unitary approach V C ū(p )γ µ u(p)(k µ + k µ) C(k + k ) V: contact interaction, momentum-dependent terms neglected S-wave Resonances are dynamically generated Dynamical coupled-channels model On-shell & off-shell effects considered Resonance dynamical generation allowed Unitarity&analyticity respected Examples: Jülich model, EBAC model JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

8 Dynamical model ingredients T = + X (a) T = V + V T X (b) = U + U X (d) V = + U U (c) = + (e) (a) T = F S F +X (b) T = V + V G T (c) V = f S f +U (d) X = U + U G X T: full amplitude X: non-pole amplitude U: driving term of X V: driving term of T S: dressed res. propagator S : bare res. propagator F : dressed res. vertex f : bare res. vertex = + = + X (a) (b) (a) S = S + S F G f S }{{} self energy Σ (b) F = f +XG f JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

9 Main features of Jülich dynamical π model (version 22) Channel-space: π η π ρ σ Two-body unitarity guaranteed by scattering equation Full analyticity (no on-shell factorization); resonance dynamical generation allowed Different channels & partial waves correlated through t- & u-channel interactions dynamical generation not easy; no need to model background with resonances Effective interaction based on chiral Lagrangians of Wess & Zumino, supplemented by additional terms for including, ω, η, a, σ σ & ρ fitted by ππ scattering; fitted by π scattering Renormalization of the nucleon pole: Roper: dynamically generated S 1 = S 1 Σ m = m Σ Genuine resonances: S 11 (1535), S 11 (165), S 31 (162), P 31 (191), P 13 (172), D 13 (152), P 33 (1232), D 33 (17) (all are 4-star s) JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

10 Partial wave amplitudes of π π (Solution 22) π η π ρ σ S 11(1535), S 11(165), S 31(162), P 31(191), P 13(172), D 13(152), P 33(1232), D 33(17) (all are 4-star s) Re S11 Im S Re P11 1 Im P W [MeV] Re S31 Im S31 Re P31 Im P W [MeV] Re P13 Im P13 Re D13 Im D W [MeV] Re D33 Im D W [MeV] JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31 Re P33 Im P

11 ew developments: inclusion of ΛK & ΣK channels K Λ, Σ Λ, Σ K Σ K Search for missing resonances t- & u-channel: SU(3) symmetry K Σ, Σ Λ Strong constraints from simultaneous fit of π π & π KY π π π Start from π + p Σ + K + : pure isospin 3/2 High spin resonances: J = 5/2, 7/2 K σ, ω, φ Λ K ρ Σ K σ, ω φ, ρ Σ K Λ K Λ K Σ JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

12 Differential cross sections of π + p Σ + K z = 1822 MeV 1845 MeV 187 MeV 1891 MeV 6 4 dσ/dω [ µb / sr ] MeV 1939 MeV 197 MeV 1985 MeV cosθ 4-star F 35(195) & F 37(195), & 3-star P 33(192) & D 35(193) needed JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

13 Polarizations of π + p Σ + K z = 1822 MeV 1845 MeV 187 MeV 1891 MeV P MeV 1939 MeV 197 MeV 1985 MeV cosθ 4-star F 35(195) & F 37(195), & 3-star P 33(192) & D 35(193) needed JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

14 Total cross sections of π + p Σ + K + S31 σ [mb].1 P33 F37.1 D33 P31 F35 D z [MeV] 4-star F 35(195) & F 37(195), & 3-star P 33(192) & D 35(193) needed JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

15 π π amplitudes (simultaneous fit to π + p Σ + K + ) Re P33 Re D Im D33 Im P33 Inclusion Σ + K + moderate improvements on π π P 33 (192) ( ): important for π + p Σ + K +, limited contributions to π π D 35 (193) ( ): similar to P 33 (192) z [MeV] z [MeV] non-polar part full result solution 22 ΣK is needed for identifying P 33 (192) & D 35 (193) P 33(1232) (1216,96) D 33(17) (1644, 252) P 33(16) (1455,694) P 33(192) (1884, 229) JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

16 ew developments: π photoproduction To get M µ & J µ, attach a photon everywhere to ; M = M µ = M µ s + M µ u + M µ t + M µ int = + U + X U = U U = JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

17 ecessity of gauge invariance Gauge invariance is a fundamental symmetry for electromagnetic interaction. Local gauge invariance for charged Dirac fields L = ψ(id/ m)ψ 1 4 Fµν F µν, where D µ is the gauge covariant derivative, D µ µ + iea µ(x). L is invariant under ψ(x) e iα(x) ψ(x), A µ A µ 1 e µα(x). The electromagnetic field is a consequence of the local gauge invariance of the Lagrangian for charged Dirac fields Local gauge invariance is a basic principle to introduce the interactions of charged particles and the electromagnetic fields Gauge invariance must be satisfied for photo-& electro-production JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

18 Ward-Takahashi Identity & current conservation Equivalent expressions for gauge invariance: GCD, WTI, GWTI Gauge invariance current conservation Gauge covariant derivative (GCD) minimum coupling µ D µ µ + iea µ(x) Ward-Takahashi Identity (WTI) for Γ µ & Γµ π k µγ µ (p, p) = S 1 p Q Q Sp 1 k µγ µ π(q, q) = 1 q Q π Q π 1 q Generalized Ward-Takahashi Identity (GWTI) for π production current k µm µ = F sτ S p+kq is 1 p + S 1 p Q f S p k F uτ + 1 p p +k Qπ p p Ftτ on shell Current conservation k µm µ = JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

19 Gauge invariance violation In a full theory (no form factors & truncations), gauge invariance is respected (minimum coupling, µ D µ µ + iea µ(x)) Real-world calculations require form factors & truncations M = U = = + + U X U Inclusion sideways form factors will destroy gauge invariance, since form factors are usually functions of the momenta of exchanged particles Truncations usually also destroy gauge invariance Prescriptions are needed to restore gauge invariance The vast majority of existing models does not satisfy gauge invariance Our model is gauge invariant (amplitudes constructed to satisfy the GWTI) JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

20 Prescription to restore gauge invariance M = M µ = M µ s + M µ u + M µ t + M µ int Generalized Ward-Takahashi Identity (GWTI) for M µ k µm µ = F sτ S p+kq is 1 p + S 1 p Q f S p k F uτ + 1 p p +k Qπ p p Ftτ Constraints on interaction current M µ int k µm µ int = Fsτ Qi + Qf Fuτ +Qπ Ftτ = + U + X U M µ int = M µ c + X G (M µ u + Mµ t + M µ c ) T Constraints on generalized contact current M µ c k µm µ c = F sτ Q i + Q f F uτ +Q π F tτ JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

21 Choosing the generalized contact current M µ c Constraints: gauge invariance; contact term; crossing symmetry Choosing the generalized contact current M c µ as {[ M c µ = g πγ 5 λ+(1 λ) q/ ] } C µ +(1 λ) γµ 2m 2m eπft C µ (2q k) µ ( (2p = e π t q 2 f t ˆF )+e k) µ f u p 2 ( ˆF = 1 ĥ(1 δs fs)(1 δu fu)(1 δt ft) (2p+k) f u ˆF )+e µ ( ) i s p 2 f s ˆF k, p, q, p : 4-momenta for incoming γ, & outgoing π, ĥ: fit parameter f x: form factors for corresponding channels Check gauge invariance: M µ c satisfies k µm µ c = F s e i + F u e f + F t e π If no form factors, i.e. f x = 1, C µ =, M c µ g πγ 5 (1 λ) γµ eπ (Kroll-Ruderman term) 2m JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

22 Photoproduction amplitudes: summary M = + B + X B T (a) M µ = F S J µ + B µ + X G B µ T M = + B + T B T (b) M µ = F S J µ s + Bµ + T G B µ T B = + + (c) B µ = M µ u + M µ t + M µ c = + T + T + T (a) J µ = J µ s + F G B µ T = + + L + L + L J µ s (b) = J µ + mµ KR G F + f G B µ L For details about J µ s, see: H. Haberzettl, F. Huang, and K. akayama, arxiv: JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

23 Channels & interaction diagrams π γ π,,r γ +, π (a) γ π,ρ,ω,a 1 (b) (a): s-channel for γ π S 11(1535) S 11(165) S 31(162) P 31(191) P 13(172) D 13(152) P 33(1232) D 33(17) (b): u- and t-channel for γ π η γ + η ρ,ω γ (c) (c): u- and t-channel for γ η (d): u- and t-channel for γ π π γ +, π π,ρ γ (d) Diagrams (c) and (d) only contribute within the FSI loop integral. JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

24 Results: dσ/dω & Σ γ for γ + p π + + n + n) ( b/sr) d /d ( p (25, 1162) 6 12 (28, 1186) (32, 1217) (6, 1416) (65, 1449) (725, 1497) (775, 1528) (825, 1558) (875, 1588) (925, 1617) (975, 1646) (deg.) (36, 1247) (4, 1277) (45, 1313) (5, 1349) (55, 1383) n) ( p (244, 1157) 6 12 (275, 1182) (322, 1219) (6, 1416) (65, 1449) (688, 1474) (75, 1513) (8, 1543) (848, 1572) (9, 163) (952, 1633) (deg.) (35, 124) (4, 1277) (45, 1313) (5, 1349) (55, 1383) Differential cross sections for γ+ p π + + n Photon spin asymmetries for γ+ p π + + n S 11 (1535), S 11 (165), S 31 (162), P 31 (191), P 13 (172), D 13 (152), P 33 (1232), D 33 (17) JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

25 Results: dσ/dω & Σ γ for γ + p π + p p) ( b/sr) d /d ( p (245, 1158) (597, 1414) (652, 145) (75, 1484) (758, 1517) (88, 1548) (859, 1579) (99, 168) (957, 1636) 6 12 (278, 1184) (32, 1217) (deg.) (36, 1247) (4, 1277) (45, 1313) (51, 1356) (555, 1386) p) ( p (244, 1157) 6 12 (275, 1182) (322, 1219) (615, 1426) (666, 1459) (73, 1483) (769, 1524) (81, 1544) (862, 158) (895, 16) (961, 1638) (deg.) (35, 124) (4, 1277) (45, 1313) (5, 1349) (551, 1384) Differential cross sections for γ + p π + p Photon spin asymmetries for γ + p π + p S 11 (1535), S 11 (165), S 31 (162), P 31 (191), P 13 (172), D 13 (152), P 33 (1232), D 33 (17) JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

26 - Results: dσ/dω & Σ γ for γ + n π + p - p) ( b/sr) d /d ( n (24, 1155) 6 12 (27, 1179) (313, 1213) (39, 1271) (436, 135) (51, 1351) (6, 1418) (64, 1444) 6 12 (36, 1249) (54, 1378) (7, 1483) (748, 1513) (8, 1545) (85, 1575) (9, 164) (95, 1633) (deg.) p) ( n (25, 1163) 6 12 (3, 123) (33, 1226) (67, 1463) (71, 1489) (75, 1514) (79, 1539) (819, 1557) (85, 1575) (9, 164) (947, 1632) (deg.) (35, 1241) (4, 1278) (45, 1315) (5, 135) (63, 1438) Differential cross sections for γ+ n π + p Photon spin asymmetries for γ+ n π + p S 11 (1535), S 11 (165), S 31 (162), P 31 (191), P 13 (172), D 13 (152), P 33 (1232), D 33 (17) JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

27 Low-energy predictions - + d /d ( p n) ( b/sr) W = 18 W = 1112 p) ( b/sr) d /d ( p W = 184 W = 1112 p) ( b/sr) d /d ( n W = 185 W = 1112 M π th = 177 M π+ n th = 179 M π p th = 173 M π p th = (deg.) (deg.) (deg.) Close to threshold: ChPT constraints for theoretical models We use Wess & Zumino chiral Lagrangians, but have form factors : isospin basis overall agreement reasonable : particle basis remarkable agreement Apart from isospin symmetry violation effects, our model works quite well at low energies A closer comparison with low-energy data requires a calculation in fully particle basis JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

28 - Contributions from loop integral & M µ c (28, 1186) 3 2 (278, 1184) (27, 1179) + n) ( b/sr) 12 2 (6, 1416) p) ( b/sr) 1 4 (597, 1414) p) ( b/sr) (6, 1418) M µ = F S J s µ + M u µ + Mµ t + M c µ ( + T G M µ u + M t µ + M c µ ) T d /d ( p 1 2 (825, 1558) d /d ( p 2 4 (88, 1548) d /d ( n 12 2 (8, 1545) : full calculation : no loop integral : no M c µ apart from K.R (deg.) (deg.) (deg.) Contribution from the loop integral is important The terms apart from the Kroll-Ruderman term in M c µ keeping gauge invariance is important give significant effects JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

29 Results: γ π total cross sections (mb) p p n W (GeV) n p p : full calculation : no loop integral : no M µ c apart from Kroll-Ruderman term Data not included in the fit Contribution from the loop integral is important Effect of the terms apart from Kroll-Ruderman term in M c µ is significant for γp π + n For γp π p, the effect of M c µ on dσ/dω is largely suppressed at backward angles by sin θ JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

30 Summary & perspectives Jülich dynamical coupled-channel model π η π ρ σ; analyticity&unitarity respected Wess & Zumino chiral Lagrangian+, ω, η, a, σ ucleon pole & dynamically generated Roper S 11 (1535), S 11 (165), S 31 (162), P 31 (191), P 13 (172), D 13 (152), P 33 (1232), D 33 (17) ew developments: π + p K + Σ + SU(3) for t- & u-channel interactions Simultaneous fit of π π & π + p Σ + K + 4-star F 35(195) & F 37(195), & 3-star P 33(192) & D 35(193) needed dσ/dω & P described quite well ew developments: π photoproduction Gauge invariance strictly respected dσ/dω & Σ γ described quite well Good low energy predictions Loop integral& M c µ (apart from K.R.) are important ext step work: ΛK & ΣK (I = 1/2) channels,ω channel Photoproduction of η, K, ω & electroproduction of π Electromagnetic couplings JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31

31 Covariance & 3-D integral equation Jülich π model TOPT T TO (p, p; s) = V TO (p, p; s) + d 3 p V TO (p, p ; s) G TO (p, s)t TO (p, p; s) G TO (p, s) = 1 s E(p ) ω(p )+i Converting to a covariant 3-D reduction like equation V(p, p; s) (2π) 3 2E(p ) 2ω(p ) 2E(p) 2ω(p) V TO (p, p; s) T(p, p; s) (2π) 3 2E(p ) 2ω(p ) 2E(p) 2ω(p) T TO (p, p; s) T(p, p; s) = V(p, p; d 3 p s) + (2π) 3 V(p, p ; s) G (p, s) T(p, p; s) G (p, s) 1 1 2E(p ) 2ω(p ) s E(p ) ω(p )+i Similarly, make 3-D reduction of the covariant photoproduction equation JLab, ewport ews, Virginia, USA π scattering& π photoproduction April 25, / 31